We compute the Drinfel'd double for the bicrossproduct multiplier Hopf algebra A = k[G] \rtimes K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] \rtimes K(H) if there is a factor-reversing group isomorphism.